Low rank positive partial transpose states and their relation to product vectors

It is known that entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. In a previous paper we presented a classification of rank four entangled PPT states which we believe to be complete. In the present paper we continue our investigations of the low rank entangled PPT states. We use perturbation theory in order to construct rank five entangled PPT states close to the known rank four states, and in order to compute dimensions and study the geometry of surfaces of low rank PPT states. We exploit the close connection between low rank PPT states and product vectors. In particular, we show how to reconstruct a PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.Comment: 29 pages, 4 figure

Extension and approximation of mm-subharmonic functions

Let $\Omega\subset \mathbb C^n$ be a bounded domain, and let $f$ be a real-valued function defined on the whole topological boundary $\partial \Omega$. The aim of this paper is to find a characterization of the functions $f$ which can be extended to the inside to a $m$-subharmonic function under suitable assumptions on $\Omega$. We shall do so by using a function algebraic approach with focus on $m$-subharmonic functions defined on compact sets. We end this note with some remarks on approximation of $m$-subharmonic functions

Spin Glasses: Model systems for non-equilibrium dynamics

Spin glasses are frustrated magnetic systems due to a random distribution of ferro- and antiferromagnetic interactions. An experimental three dimensional (3d) spin glass exhibits a second order phase transition to a low temperature spin glass phase regardless of the spin dimensionality. In addition, the low temperature phase of Ising and Heisenberg spin glasses exhibits similar non-equilibrium dynamics and an infinitely slow approach towards a thermodynamic equilibrium state. There are however significant differences in the detailed character of the dynamics as to memory and rejuvenation phenomena and the influence of critical dynamics on the behaviour. In this article, some aspects of the non-equilibrium dynamics of an Ising and a Heisenberg spin glass are briefly reviewed and some comparisons are made to other glassy systems that exhibit magnetic non-equilibrium dynamics.Comment: To appear in J. Phys.: Condens. Matter, Proceedings from HFM2003, Grenobl

Production and verification of the first Atlantic salmon (Salmo salar L.) clonal lines

In several fish species homozygous and heterozygous clonal lines have been produced using gynogenetic and androgenetic techniques. These lines are standardized and can be reproduced over generations. In rainbow trout such lines have existed for decades and has become important research tools in genome studies as well as in studies of commercially important traits. The Atlantic salmon is one of the best studied fish species globally, but all experiments are done on fish of wild or domesticated origin and access to standardized immortal fish lines would be of great benefit. Here, we describe the protocols developed to produce mitotic gynogenes, and from these the first clonal lines in Atlantic salmon.publishedVersio